Another Fork in the Road
Continuing on your journey to see the wizard, you come to another fork in the road. At this juncture, stand 3 men ... all looking wise and wizardly. Again, there is a sign ... it reads:
ONE OF US TELLS THE TRUTH, ONE OF US LIES, AND ONE OF US MAY
DO EITHER
(each of us knows which one is truthful, which one is the liar, and which one waffles between the truth and a lie .......... travelers may ask one of us, and only one of us, one question whose answer is either "yes" or "no")
What question would you ask to determine which of the roads you select, to continue your journey?
ONE OF US TELLS THE TRUTH, ONE OF US LIES, AND ONE OF US MAY
DO EITHER
(each of us knows which one is truthful, which one is the liar, and which one waffles between the truth and a lie .......... travelers may ask one of us, and only one of us, one question whose answer is either "yes" or "no")
What question would you ask to determine which of the roads you select, to continue your journey?
posted by Zaux at
5:55 AM
26 Comments:
it doesn't seem that this can be done with only one question
what if you were talking to the yes/no guy,then his answer means nothing,no matter how fancy the question is
hi Knightmare ....
there is a question which works ... although, when I first read the question, I had the same response
"which way do I need to go?"
The sign was clearly written by a liar.
Will the person to the right of you say to turn right?
I think there is more than one way of doing it, but I'll give this one:
Let T, L and R be truthful, liar and random.
As any of them, "What would you say if I was to ask you, in your
current mental state, 'is road 1 the correct road?'?". If the
answer is "yes", then you want road 1. If the answer is no, then
you want road 2,
Explanation: If road 1 is the correct road.
If I asked T is road 1 the correct road, he'd say "yes".
If I asked L is road 1 the correct road, he'd say "no".
If road 2 is the correct road.
If I asked T is road 1 the correct road, he'd say "no".
If I asked L is road 1 the correct road, he'd say "yes".
If road 1 is the correct road.
As any of them, "What would you say if I was to ask you, in your
current mental state, 'is road 1 the correct road?'?".
T: as he would have said "yes", he tells the truth and says "yes".
L: as he would have said "no", he lies and says "yes".
R: behaves as T or L, so says "yes".
If road 2 is the correct road.
As any of them, "What would you say if I was to ask you, in your
current mental state, 'is road 1 the correct road?'?".
T: as he would have said "no", T truthfully says that he would have said "no".
L: as he qould have said "yes" he lies and says "no".
R: behaves as T or L, so says "no".
The "in your current mental state" is crucial. It is redundant if
you happen to be asking T or L though.
the sign is accurate ...
no correct solution yet ... unless of course, you can logically explain why your solution works ...:)
Hi Chris ... cross posts
your question seems to work ...
before I reveal the published question, I want to get other responses to see if there are other approaches which work
I assume that Zaux, hadn't seen my solution when he posted. I haven't used the fact that each knows who each other are. Tht still leaves (I'm sure a soluble problem) that it isn't clear if the know random's current mental state.
I have assumed that random has a mental state, is aware of it and so forth; i.e. he doesn't simply say "yes" or no" entirely randomly and independently of what you say to him.
This one is mind boggling.. Pass!
I have been thinking about this, I think I have the solution! Give me a minute to write it logically!
"What would you not say to 'is Road a good?'"
Truth teller:
if A is good:
would say yes to 'is Road A good'
would not say no
tells truth and says no
Liar:
If A is good:
would say no to 'is Road A good'
would not say yes
lies and says no
Random:
Acts as truth teller or liar. See above
Truth teller:
if A is bad:
would say no to 'is Road A good'
would not say yes
tells truth and says yes
Liar:
If A is bad:
would say yes to 'is Road A good'
would not say no
lies and says yes
Random:
Acts as truth teller or liar. See above
Yes=Road A is bad
No=Road A is good.
Cam
I think the question you should ask is this:
"If I were to ask you 'Is the left road the correct route?' would you answer 'Yes' ?"
Notice that this is a question about a question. Let's call the whole thing question A and the part 'Is the left road the correct route?' question B.
First, let us assume that the left road is the correct road.
If you asked the truth teller question A, he would say Yes.
If you asked the liar question A he would say No.
If you asked the waffler question A he could say either Yes (in truth mode) or No (in lie mode).
Now, The truth teller would always answer Yes to question B so his answer to question A would be Yes.
The liar would always answer No to question B so he would lie about question A and say Yes.
If the waffler were in truth mode he would act exactly like the truth teller and say Yes.
If the waffler were in lie mode he would act exactly like the liar and say Yes.
Now, assume the right-hand road is the correct road.
If you asked the truth teller question A, he would say No.
If you asked the liar question A, he would say Yes.
If you asked the waffler question A he would say No (in truth mode) or Yes (in lie mode).
In this case, the truth teller says No to question B, making his truthful answer to question A "No".
The liar would say Yes to question B so he lies about question A and says "No".
Once again, the waffler gives the truth-teller's answer if he is in truth mode - "No"
And if the waffler is in lie mode he will give the liar's answer, "No".
So, If you asked this question and got the answer "Yes", then take the left road.
If you get the answer "No" then take the right road.
Does XOR(Road A=good road,You are telling lies now=TRUE)=TRUE ?
XOR=exclusive Or
XOR(A,B) evaluates as true if A or B is true, but false if A and B is true
if Road A=good
Truth teller:
(Road A=good road)=TRUE
(You are telling lies now=TRUE)=FALSE
XOR(TRUE ,FALSE)=TRUE
Truth teller says truth and says yes
If Road A=good
Liar:
(Road A=good road)=TRUE
(You are telling lies now=TRUE)=TRUE
XOR(TRUE, TRUE)=FALSE
Liar lies and says yes
Random acts as Truth teller or Liar as above
if Road A=bad
Truth teller:
(Road A=good road)=FALSE
(You are telling lies now=TRUE)=FALSE
XOR(FALSE,FALSE)=FALSE
Truth teller says truth and says no
If Road A=bad
Liar:
(Road A=good road)=FALSE
(You are telling lies now=TRUE)=TRUE
XOR(FALSE,TRUE)=TRUE
Liar lies and says no
Random acts as Truth teller or Liar as above
Cam
hey guys ... I will be gone most of the afternoon, but will return to post a published solution ... and to enjoy everyone's analysis
Sorry I just re-read my solution and I seem to have got myself confused with my As and Bs!!
Please see below for corrected version!
I think the question you should ask is this:
"If I were to ask you 'Is the left road the correct route?' would you answer 'Yes' ?"
Notice that this is a question about a question. Let's call the whole thing question A and the part 'Is the left road the correct route?' question B.
First, let us assume that the left road is the correct road.
If you asked the truth teller question B, he would say Yes.
If you asked the liar question B he would say No.
If you asked the waffler question B he could say either Yes (in truth mode) or No (in lie mode).
Now, The truth teller would always answer Yes to question B so his answer to question A would be Yes.
The liar would always answer No to question B so he would lie about question A and say Yes.
If the waffler were in truth mode he would act exactly like the truth teller and say Yes.
If the waffler were in lie mode he would act exactly like the liar and say Yes.
Now, assume the right-hand road is the correct road.
If you asked the truth teller question B, he would say No.
If you asked the liar question B, he would say Yes.
If you asked the waffler question B he would say No (in truth mode) or Yes (in lie mode).
In this case, the truth teller says No to question B, making his truthful answer to question A "No".
The liar would say Yes to question B so he lies about question A and says "No".
Once again, the waffler gives the truth-teller's answer if he is in truth mode - "No"
And if the waffler is in lie mode he will give the liar's answer, "No".
So, If you asked this question and got the answer "Yes", then take the left road.
If you get the answer "No" then take the right road.
The published solution does provide a rather cool question:
Select one of the three men standing at the fork and ask him:
"If I were to ask you whether the left fork leads to where I'm going, and you chose to answer that question with same degree of truth as you answer this question, would you then answer 'yes'?"
The truth teller will say "yes" if the fork leads to where you're going and "no" otherwise. The liar will answer the same, since he will lie about where the left fork leads, and he will lie about lying. The third man may either lie or tell the truth about this one question, but either way he is behaving like either the truthteller or the liar and thus must correctly tell you the road to your destination.
Zaux. That's what I said. Your "with the same degree of truth" is the same as my "in your current mental state".
I still suspect that there is another solution.
"If you believed that an incorrect answer would lead to your death, would you tell me the left fork leads where I am going?" Said while holding a knife to the man's throat.
Hi Simon. That's a quite interesting solution. But, I think that
Random cannot answer the question. He'll to say to himself, "if
you had asked the inner question, my answer would have depended on
me being in T or L mode when you asked it". So he can't answer
uniquely; he'd explode in a metaphysical dichotomy.
But I also, see that he might actually answer the complete
question in the same mode. We don't know enough about the nature
of Random to safely make that assumption. So you must add the "in
your current mental state" to eliminate that possibility.
If Random doesn't behave rationally, then I suspect that it isn't
possible to solve the problem as given.
If it's any consolation, I had had to think quite hard before I
felt brave enough to challenge your solution.
The question I proposed can be simplified to "If I were to ask you, right now, "Is road 1 the correct road?', what would you answer?"
Simon. I hope that you take my nit-picking as a sign of respect. The fundamental reason that the proposed solution works, is because of ensuring that Random answers the inner and out question entirely as T or L. That fact should be made explicitly clear.
This question is addressed to those who have posted a "correct" solution i.e. Chris Simon H and anonymous.
Suppose that I am one of the three people that you can ask.
Also, let's suppose that road 1 is the correct road and I know this.
You ask me your question. I reply: no. (and the universe doesn't implode in the presence of a metaphysical impossibility)
Since you have both argued that a truth teller, a liar and a randomer would all say yes, what is the status of my answer?
Hi Cam. I only just noticed your post. I'm sure that you accept
that R being in a guaranteed state throughout as being crucial.
The real reason for posting is curiosity about why you posed the
question in the negative. Why didn't you use "what would you say
to 'is road a good?'"
Hi Mr F. You cannot be one of the three people referred to in the question. So the status of your answer is 42.
Mr F, your status has been changed to, "for the X-files".
Chris,
I concur, R being in the current guaranteed state is crucial, but my thoughts are its implied (if not Random could only yield "I know. If he's lying" or "I don't know. If he's telling the truth" as an answer. Since neither of these are yes or no, he can't answer, thus if it is not implied, and the question yields only silence, then you found the Random.)
The XOR statement completely avoids this issue by asking if they are currently lying.
As far as why I posed the question in the negative, paraphrasing-> why NOT? Answer: Why not ;)
Seriously, no reason. Just to be different. The logic is identical.
Cam
Chris ... your "in your current mental state" comment, for some reason, did not equate in my stupid brain to being logically equivalent to "with the same degree of truth" comment. Sorry, I do now understand it. For some reason, it just did not register... :)
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